ANT 07 Homework 2
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\begin{center}
\Large\bf Homework 2 for Math 581F, Due FRIDAY October 12, 2007\end{center}
Each problem has equal weight, and parts of problems are worth the
same amount as each other.
\begin{enumerate}
\item Let $\vphi:R\to S$ be a homomorphism of (commutative) rings.
\begin{enumerate}
\item Prove that if $I\subset S$ is an ideal, then $\vphi^{-1}(I)$
is an ideal of~$R$.
\item Prove moreover that if $I$ is prime, then $\vphi^{-1}(I)$ is
also prime.
\end{enumerate}
\item Let $\O_K$ be the ring of integers of a number field.
The Zariski topology on the set $X=\Spec(\O_K)$ of all prime ideals
of $\O_K$ has closed sets the sets of the form
$$
V(I) = \{ \p\in X : \p \mid I\},
$$
where~$I$ varies through {\em all} ideals of $\O_K$, and $\p\mid I$
means that $I \subset \p$.
\begin{enumerate}
\item Prove that the collection of closed sets of the
form $V(I)$ is a topology on $X$.
\item Prove that the conclusion of (a) is still true if $\O_K$ is replaced
by an order in $\O_K$, i.e., a subring that has finite
index in $\O_K$ as a $\Z$-module.
\end{enumerate}
\item Let $\alpha = \sqrt{2} + \frac{1+\sqrt{5}}{2}$.
\begin{enumerate}
\item Is $\alpha$ an algebraic integer?
\item Explicitly write down the minimal polynomial of $\alpha$
as an element of $\QQ[x]$.
\end{enumerate}
\item Which are the following rings are orders in the given
number field.
\begin{enumerate}
\item The ring $R = \ZZ[i]$ in the number field $\QQ(i)$.
\item The ring $R = \ZZ[i/2]$ in the number field $\QQ(i)$.
\item The ring $R = \ZZ[17i]$ in the number field $\QQ(i)$.
\item The ring $R = \ZZ[i]$ in the number field $\QQ(\sqrt[4]{-1})$.
\end{enumerate}
\item Give an example of each of the following, with proof:
\begin{enumerate}
\item A non-principal ideal in a ring.
\item A module that is not finitely generated.
\item The ring of integers of a number field of degree~$3$.
\item An order in the ring of integers of a number field of degree~$5$.
\item A non-diagonal matrix of left multiplication by an element
of~$K$, where~$K$ is a degree~$3$ number field.
\item An integral domain that is not integrally closed in its field of fractions.
\item A Dedekind domain with finite cardinality.
\item A fractional ideal of the ring of integers of a number
field that is not an integral ideal.
\end{enumerate}
\end{enumerate}
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